A differential measures the RATE of change of something whereas filling in values measures the actual change. For example your first example states that the change from x=10 to x=11 is 21 by filling in values but your integral is calculated incorrectly. It should read.

which is your original fitting in of values and is perfectly accurate. Differentials only really come into play when modelling things that do not have a linear relationship with some reference frame or other quantity, such as the density of the sun which varies with radius or heat capacities which vary with temperature.

Ok. Well again in your previous example the use of the derivative on [tex] x^2 [/tex] would tell you the rate of changeof the function. Say [tex] x^2 [/tex] represented a change of position with time, then the derivative would tell us what speed a particle was travelling at at a certain point on the graph. so if you changed position from 10 meters to 11 meters the corresponding difference in the derivative would tell you how much your speed changed in that interval as opposed to the rate of change at a point. so a value of 2 just happens to be the acceleration if you took the second derivative which is constant throughout but thats because you took a unit change.

But writing a couple of lines is not necessarily less computationally intensive than writing 5 lines if the question is significantly different. Once you've computed the differential stuff it becomes very easy to work it out for many different values of dx, and if they are sufficently small this is all you require. JUst imagine cases where the derivative is easier than the original function to calculate! Say I want you to work out an estimate for changing 4x^5+x^4 at x=1 f dx=0.1, 0.01, 0.0023. Wouldn't it be easier touse the linearizations that actually compute the damn function at all those points? Suppose I want to charge you a million dollars per floating point operation, which would you rather use then? Remember, the SIMPLIFIED situations you are using things for aren't all there is. If you weren't using linearization all over the place you wouldn't be able to to calculate anything worth a damn.

I agree that soemtimes I feel the only reason we teach the use of differentials for approximating differences is to introduce the concept of differentials. I.e. that approximation technique is not really the best use of them.

More important is the method of substitution, where you substitute x = x(u) and dx = (dx/du) du. I.e. differentials are a convenient calculating tool in many situations, such as path integrals where the general integrand looks like A(x,y)dx + B(x,y)dy, and the differentials are visible in there.

Also differential equations are called that for a reason. For instance solving them by separation of variables is another important use of differentials, where you solve